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1 007 I II III 1,, 3, 4, 5, 6, ε-n 1

2 ε-n ε-n ε-n. {a } =1 a ε N N a a <ε ε N [ >N= a a <ε] ε N 1

3 3. {a } =1 a a {a } =1 lim a = a a a ( ) = , 0.99, 0.999, xy (1,a 1 ), (,a ), (3,a 3 ), 1 ( 1,a 1 ), ( 1/,a ), ( 1/3,a 3 ),... y 1

4 4 a a a 3 a 4 a : y N x = 1/N ±ε ε a ε N : N a ε lim a = a 1,, 6

5 5 3 /m ( 0) +1 4, 5 1/ 0 1. {a } =1 a = 1 lim a =0. r r < 1 a = r lim a =0 3. a a = a! lim a =0 4. {a } =1 {b } =1 {c } =1 c m 1 = a m, {a } =1 {b } =1 {c } =1 c c m = b m c 5. {a } =1 a {b } =1 b c = a + b {c } =1 a + b 6. {a } =1 {b } =1 b = a 1 + a + + a {a } =1 a {b } =1 a

6 6 {a } =1 1,, 3 4, 5, 6 {a } =1 a a {a } =1 ε ε N N a a <ε ε ε N a a <ε N ε ε N ε N

7 7 N ε N [ >N= a a <ε] ε >0 N N N ε N N ε N N ε 1 1 ε N 1 0 <ε N [ ] ε N >N= 1 0 <ε

8 8 ε ε ε N ε ε N ε =0.1 N ε =0.01 N ε ε N ε N ε N ε ε ε 0 ε ε 0 ε ε 0 ε ε ε 0 [ >N= a a <ε 0 ] N N ε 0 N {a } =1 a =1/, a =0 a a <ε 0 1 <ε 0 > 1 ε 0

9 9 a 0 <ε 0 N 1/ε 0 N N 1 ε 0! N N ε ε ε N. ε 1/ε N N 1 < 1 N ε 1 0 = 1 1 lim =0 ε N [ >N= a a <ε] N ε N >N= a a <ε a x +3=a x

10 10 a x [x +3=a] x = a 3 x +3=(a 3) + 3 = a x x = a 3 a x +3=a x 1 ε ε ε 0 ε 0 N a 0 <ε 0 N ε 0 {a } =1 N {a } =1 a 0 = r 0 = r = r r 1 r <ε 0 r =0 r =0<ε 0 N r 0 0 < r < 1 1 r s r = =1+s (s >0) 1 (1 + s) = 1 1+s + ( 1) s + + s 1 + s

11 11 s>0 (1 + s) =1+s + ( 1) s + + s 1 + >s r < 1 s 1 s <ε 0 r <ε 0 N N> 1 sε 0 N ε N ε ε N {a } =1 {a } =1 N s N N N. ε r =0 a =0 a 0 =0<ε lim a =0 r 0 s 1 r =1+s

12 1 N N 1 sε N s (1 + s) =1+s + N a 0 = r = ( 1) s + + s 1 + s >s 1 (1 + s) < 1 s < 1 Ns 1 1 sε s = ε lim a =0 3 3 a =0 a 0 ε ε 0 ε 0 N a 0 =0<ε 0 N ε 0 N {a } =1 {a } =1 N a 0 = a! = a! a! <ε 0 ε 0 a a! = a a 1 a a 1

13 13 a M M ( ) a M < a M a! M! M ( ) a M M a <ε 0 M! M a M a M /M! M ( a /M ) M ( ) a < M! M M ε M 0 (1) N N a /M < 1 ( a /M ) M >N (1) N N ε 0 r < 1 lim r =0 N (1) N lim r =0 N N. a M a /M < 1 ( ) a lim =0 M

14 14 ε N N ( ) a M! M M ε M M a 0 = a! < a M M! ( ) M a = M M M M! ( ) a < M M M! M M! M ε = ε M lim a =0 lim ( a /M ) =0 ε N [ >N= ( ) a <ε] M ε N [ >N= ( ) a < M! ] M M ε M? ε N [ >N= ( ) a <ε] M ε M!ε/M M? ] ε N [ >N= a <ε ()!

15 15 ε N ε N [ >N= ( ) a <ε] M ε N 3 1. () ε ε 0. ε 0 M!/M M (3) ε 3. (3). ε N 4. N () N > N = a /! <ε 0 4, 5, 6 (3) a! < a M M! ( ) M a M ε-n?

16 16 4 1,, 3 N {a } =1 N 3 {a } =1 ε ε c {b } = ε ε c {c } = ε ε c 3: {a } =1 {b } =1? 1,, 3 ε-n ε N [ >N= c c <ε] (4) ε N ε N [ >N= a c <ε] (5)

17 17 ε N [ >N= b c <ε] (6) (5) (6) ε ε (4) ε (4) ε ε 0 ε ε 0 (5),(6) ε N N N N c c <ε 0 N c c <ε(5) (6) c = a +1 a +1 c <ε 0 (5) (5) ε ε 0 (5) N N a +1 >N a c = b b c <ε 0 (6) (6) ε ε 0 (6) N N b >N b

18 18 c c <ε 0 [ >N a 1 ] [ >N b ] N >N c c <ε 0 N >N a 1 >N b c c <ε 0 N a 1 N b N N = max{n a 1, N b } N N N a 1 N b N a N b. ε {a } =1 {b } =1 c N a N a a c <ε N b N b b c <ε N a N b N N =N N +1 > N +1 = N +1 > N = N = N N b >N N a c c = a +1 c <ε c c = b c <ε c c <ε lim c = c

19 {a } = N a ε ε a ε {b } = N b ε b 4: {a } =1 {b } =1 5 a 1 + b 1 ε a + b ε : {a + b } =1 ε a a b b ε a + b a + b ε a + b a + b ε a a b b ε/

20 0 a + b a + b ε ε ε N [ >N= (a + b ) (a + b) <ε] ε N [ >N= a a <ε] ε N [ >N= b b <ε] ε ε 0 ε 0 >N (a + b ) (a + b) <ε 0 N (a + b ) (a + b) <ε 0 a + b a + b (a + b ) (a + b) = (a a)+(b b) a a + b b a a + b b <ε 0 (7) a a b b (7) (7) (7) a a b b ε 0 ε 0 ε 0 a a < ε 0 b b < ε 0

21 1 a a <ε 0 / ε ε 0 / N N a >N a b b <ε 0 / ε ε 0 / N N b >N b N a N b a a <ε 0 / b b <ε 0 / a a + b b <ε 0 (a + b ) (a + b) <ε 0 N N a N b max{n a,n b } N (a + b ) (a + b) <ε 0. ε lim a = a lim b = b ε N a N b >N a a a <ε/ >N b b b <ε/ N a N b N >N a a <ε/ b b <ε/ (a a)+(b b) a a + b b < ε + ε = ε >N (a + b ) (a + b) <ε a a <ε 0 / b b <ε 0 / ε 0 ε 0 /

22 a a <ε 0 b b <ε 0 (a + b ) (a + b) a a + b b < ε 0 < ε 0 < ε 0 ε 0 a a b b ε 0 ε 0 / < ε 0 < ε 0. 0 {a } =1 {b } =1 a b lim (a + b )=a + b, lim (a b )=a b, lim (a b )=ab, a lim = a b b b b {a } =1 a a a a a a a

23 3 [ ] ε N >N= a 1 + a + + a a <ε ε N [ >N= a a <ε] ε ε 0 ε 0 N a 1 + a + + a a <ε 0 N a k a a 1 + a + + a k a 1 + a + + a a = a 1 + a + + a a = (a 1 a)+(a a)+ +(a a) 1 a 1 + a + + a a a 1 a + a a + + a a ε 0 a 1 a a a ε 0 ε 0 a k a <ε 0 k k a k a k a k a ε 0

24 4 a k a <ε 0 / a k a <ε 0 a k a <ε 0 a k a ε 0 ε ε 0 N N a a 1 a < a 1 a + + a N a a + + a N a a = a 1 a + + a Na a + a N a+1 a + ε ε a a + N a ε 0 (8) >N a N a < =1 N a ε 0 <ε 0 N a ε 0 a 1 a + a a + + a Na a a 1 a + a a + + a Na a <ε 0 N N 1 a 1 a,..., a Na a M N N> MN a ε 0 N N a a 1 + a + + a a <ε 0 + ε 0 =ε 0

25 5 < ε 0 < ε 0 ε 0 ε ε 0 ε ε 0 ε 0 / k>n a = a k a < ε 0 N a M a 1 a,..., a N a a N > M N a ε 0 / N N N a a 1 + a + + a a < ε 0 + ε 0 = ε 0. {a } =1 a ε N a >N a a a < ε N a M a k a <M k =1,,...,N a >N a a 1 + a + + a a a 1 a + + a Na a + a N a+1 a + + a a N MN a MN a N ε + ( N a)ε

26 6 N N a N >N= a 1 + a + + a a < ε + ε = ε. (8) 4 a 1 a + + a Na a 0, N a 1 lim b a ε 0 ε 0 lim b a =0 lim b a lim b a b a [0,ε 0 ] ε 0 0 ε-n

2010 IA ε-n I 1, 2, 3, 4, 5, 6, 7, 8, ε-n 1 ε-n ε-n? {a n } n=1 1 {a n } n=1 a a {a n } n=1 ε ε N N n a n a < ε

2010 IA ε-n I 1, 2, 3, 4, 5, 6, 7, 8, ε-n 1 ε-n ε-n? {a n } n=1 1 {a n } n=1 a a {a n } n=1 ε ε N N n a n a < ε 00 IA ε-n I,, 3, 4, 5, 6, 7, 8, 9 4 6 ε-n ε-n ε-n? {a } = {a } = a a {a } = ε ε N N a a < ε ε-n ε ε N a a < ε N ε ε N ε N N ε N [ > N = a a < ε] ε > 0 N N N ε N N ε N N ε a = lim a = 0 ε-n 3 ε N 0 < ε